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Math Help - Please solve my question

  1. #1
    stevengoh911
    Guest

    Please solve my question

    Given
    x,y ∈ [0,2c]
    f(x),g(y) ∈ [0,2c]
    Show
    | xy-f(x)-g(y)|≥ c≤
    exist

    I don't really get this question.
    Please help me
    Last edited by ThePerfectHacker; July 10th 2007 at 09:38 AM.
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  2. #2
    Global Moderator

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    Quote Originally Posted by stevengoh911 View Post
    Given
    x,y ∈ [0,2c]
    f(x),g(y) ∈ [0,2c]
    Show
    | xy-f(x)-g(y)|≥ c≤
    exist
    It is assumed that c>0:

    Consider the points (0,2c),(2c,0),(0,0) if these work then we are finished. Otherwise,
    |f(0)+g(0)|<c^2
    |f(2c)+g(0)|<c^2
    |f(0)+g(2c)|<c^2

    Now, the claim is that |f(2c)+g(2c)-4c^2|\geq c^2.
    This is true because,
    f(2c)+g(2c) = [f(2c)+g(0)] + [f(0)+g(2c)] - [f(0)+g(0)] < c^2+c^2+c^2 = 3c^2
    But then,
    f(2c)+g(2c) - (2c)(2c) < 3c^2 - (2c)(2c) = -c^2
    Thus,
    |f(2c)+g(2c) - (2c)(2c)|\leq c^2
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